1. Field of the Invention
This invention relates to a method for displaying three-dimensional images using a cathode ray tube (CRT) and a flexible membrane mirror driven by a waveform. More specifically, the invention relates to an improved method of displaying three-dimensional images by driving the mirror with a combined sinusoidal, triangular waveform.
2. Description of the Prior Art
A flexible membrane mirror reflects a two-dimensional image displayed on a CRT. The flexible membrane mirror is displaced so that a surface of the mirror is alternately convex and concave. This displacement or vibration of the mirror is caused by an action of a loudspeaker which is mounted directly behind the mirror. This action of the loudspeaker moves an apparent image plane of the mirror toward and away from a viewer. As the image plane moves, two-dimensional images are displayed sequentially on the CRT screen. When the motion of the mirror and the timing of the CRT images are properly synchronized, the viewer perceives a three-dimensional image. The apparent image planes should be spaced equally in distance from each other along a viewing axis of the mirror for geometric fidelity, they each should be displayed for an equal duration of time for uniform brightness, and the mirror should operate quietly, i.e., without audible noise.
The basic technology for making such a display system is known in the art, as illustrated, for example, in U.S. Pat. No. 3,493,290 issued to Traub, as well as in a 1968 article by Traub, "A New 3-Dimensional Display Technique", distributed by the Mitre Corporation.
One conventional waveform with which to drive the mirror is a triangle wave. Since the slope of a triangle wave is constant, the mirror moves toward and away from the viewer at a constant speed, and the image plane moves toward and way from the viewer at nearly a constant speed. Therefore, if two-dimensional images are displayed on the CRT with equal spacing in time, the image planes will also be approximately equally spaced in distance along the viewing axis. Because this spacing of the image planes along the viewing axis is only approximately equal in distance, there is some geometric infidelity.
This approximate equal time spacing of the two-dimensional images has a further important advantage in that there is virtually no gap or interval of time between displayed two-dimensional images. In other words, the CRT is displaying two-dimensional images virtually at all times; the time interval between displayed two-dimensional images approaches zero and there is essentially no wasted time. Moreover, two-dimensional images can each be displayed for a long amount of time, giving a bright image.
Unfortunately, the triangle wave has a very serious shortcoming because it contains high frequency harmonics. These harmonics cause the mirror to emit an unacceptable level of sound, and also excite higher harmonics of the mirror's resonant frequency, causing distortion in the mirror surface and also in the displayed image. Because of these problems of noise and distortion, actual display systems commonly use a sine wave for the driving waveform.
A pure sine wave has no harmonics whatsoever, and hence produces undistorted mirror surfaces and also produces the least mirror noise.
However, a disadvantage of a sine wave drive signal relates to geometric fidelity. A sine wave has a constantly changing slope; therefore, to keep the perceived three-dimensional image in geometric fidelity, that is, to keep the image plane locations equally spaced in distance along the viewing axis, the two-dimensional images must be unequally spaced in time. In other words, there are gaps or intervals of time when the CRT is displaying no two-dimensional image.
Another problem with a sine wave drive signal is that image brightness is reduced. To maintain uniform brightness for all the two-dimensional images, each image must be displayed for the same amount of time as every other two-dimensional image. This amount of time can only be as long as the shortest image time, which occurs when the slope of the sine wave is at its maximum. The approximate magnitude of this reduction in brightness can be found by simply finding the ratio of the maximum slope of the sine wave to the slope of the triangle wave.
Let:
Period=T PA1 Amplitude=C PA1 w=frequency (rad/sec=2.sup..pi. /T ##EQU1## Thus it can be seen that although the sine wave gives quiet operation without distortion of the image, it also reduces the brightness of the image by a factor of 1.57. The above analysis assumes that the CRT phosphor response is linear, meaning that the phosphor brightness increases in direct proportion to the time the image is displayed. If, in fact, the phosphor response is non-linear, the brightness ratio could be greater than or less than the ratio described in the above analysis.
Moreover, because of the mirror's geometric optics, neither a triangle wave nor a sine wave results in a purely linear relationship between displacements of the mirror and positions of the perceived image along the viewing axis.
Hence, there is a need for a driving waveform which produces a bright image while operating quietly and without distortion. There is also a need for a driving waveform which produces a linear relationship between the displacement of the mirror and the position of the perceived image along the viewing axis.
Accordingly, it is a primary object of this invention to provide a method of displaying a three-dimensional image characterized by quiet operation without distortion while also yielding a bright image. It is also an object of this invention to provide a method which produces a nearly linear relationship between the displacement of the mirror and the position of the perceived image along the viewing axis.